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0. 前言
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2. 复杂度
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2.1. 算法效率评估
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2.2. 时间复杂度
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2.3. 空间复杂度
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2.4. 小结
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3. 数据结构
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3.1. 数据结构分类
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3.2. 基本数据类型
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3.3. 数字编码 *
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3.4. 字符编码 *
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3.5. 小结
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4. 数组与链表
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4.1. 数组
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4.2. 链表
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4.3. 列表
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</a>
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</li>
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<li class="md-nav__item">
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<a href="../../chapter_array_and_linkedlist/summary/" class="md-nav__link">
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4.4. 小结
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</a>
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</li>
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</ul>
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</nav>
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</li>
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<li class="md-nav__item md-nav__item--nested">
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<input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_6" >
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<div class="md-nav__link md-nav__link--index ">
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<a href="../../chapter_stack_and_queue/">5. 栈与队列</a>
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<label for="__nav_6">
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<span class="md-nav__icon md-icon"></span>
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</label>
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</div>
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<nav class="md-nav" data-md-level="1" aria-labelledby="__nav_6_label" aria-expanded="false">
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<label class="md-nav__title" for="__nav_6">
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<span class="md-nav__icon md-icon"></span>
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5. 栈与队列
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</label>
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<ul class="md-nav__list" data-md-scrollfix>
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<li class="md-nav__item">
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<a href="../../chapter_stack_and_queue/stack/" class="md-nav__link">
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5.1. 栈
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</a>
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</li>
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<li class="md-nav__item">
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<a href="../../chapter_stack_and_queue/queue/" class="md-nav__link">
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5.2. 队列
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</a>
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</li>
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<li class="md-nav__item">
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<a href="../../chapter_stack_and_queue/deque/" class="md-nav__link">
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5.3. 双向队列
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</a>
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</li>
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<li class="md-nav__item">
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<a href="../../chapter_stack_and_queue/summary/" class="md-nav__link">
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5.4. 小结
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</a>
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</li>
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</ul>
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</nav>
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</li>
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<li class="md-nav__item md-nav__item--nested">
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<input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_7" >
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<div class="md-nav__link md-nav__link--index ">
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<a href="../../chapter_hashing/">6. 散列表</a>
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<label for="__nav_7">
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<span class="md-nav__icon md-icon"></span>
|
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</label>
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</div>
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<nav class="md-nav" data-md-level="1" aria-labelledby="__nav_7_label" aria-expanded="false">
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<label class="md-nav__title" for="__nav_7">
|
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<span class="md-nav__icon md-icon"></span>
|
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6. 散列表
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</label>
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<ul class="md-nav__list" data-md-scrollfix>
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<li class="md-nav__item">
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<a href="../../chapter_hashing/hash_map/" class="md-nav__link">
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6.1. 哈希表（New）
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</a>
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</li>
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<li class="md-nav__item">
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<a href="../../chapter_hashing/hash_collision/" class="md-nav__link">
|
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6.2. 哈希冲突（New）
|
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</a>
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</li>
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<li class="md-nav__item">
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<a href="../../chapter_hashing/hash_algorithm/" class="md-nav__link">
|
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6.3. 哈希算法（New）
|
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</a>
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</li>
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<li class="md-nav__item">
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<a href="../../chapter_hashing/summary/" class="md-nav__link">
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6.4. 小结
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</a>
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</li>
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</ul>
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</nav>
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</li>
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<li class="md-nav__item md-nav__item--nested">
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<input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_8" >
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<div class="md-nav__link md-nav__link--index ">
|
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<a href="../../chapter_tree/">7. 树</a>
|
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<label for="__nav_8">
|
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<span class="md-nav__icon md-icon"></span>
|
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</label>
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</div>
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<nav class="md-nav" data-md-level="1" aria-labelledby="__nav_8_label" aria-expanded="false">
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<label class="md-nav__title" for="__nav_8">
|
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<span class="md-nav__icon md-icon"></span>
|
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7. 树
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</label>
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<ul class="md-nav__list" data-md-scrollfix>
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<li class="md-nav__item">
|
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<a href="../../chapter_tree/binary_tree/" class="md-nav__link">
|
|
|
7.1. 二叉树
|
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</a>
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</li>
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<li class="md-nav__item">
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<a href="../../chapter_tree/binary_tree_traversal/" class="md-nav__link">
|
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7.2. 二叉树遍历
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</a>
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</li>
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<li class="md-nav__item">
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<a href="../../chapter_tree/array_representation_of_tree/" class="md-nav__link">
|
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7.3. 二叉树数组表示
|
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</a>
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</li>
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<li class="md-nav__item">
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<a href="../../chapter_tree/binary_search_tree/" class="md-nav__link">
|
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7.4. 二叉搜索树
|
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</a>
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</li>
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<li class="md-nav__item">
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<a href="../../chapter_tree/avl_tree/" class="md-nav__link">
|
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7.5. AVL 树 *
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</a>
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</li>
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<li class="md-nav__item">
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<a href="../../chapter_tree/summary/" class="md-nav__link">
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7.6. 小结
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</a>
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</li>
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</ul>
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</nav>
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</li>
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<li class="md-nav__item md-nav__item--nested">
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<input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_9" >
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<div class="md-nav__link md-nav__link--index ">
|
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<a href="../../chapter_heap/">8. 堆</a>
|
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<label for="__nav_9">
|
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|
<span class="md-nav__icon md-icon"></span>
|
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</label>
|
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</div>
|
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<nav class="md-nav" data-md-level="1" aria-labelledby="__nav_9_label" aria-expanded="false">
|
|
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<label class="md-nav__title" for="__nav_9">
|
|
|
<span class="md-nav__icon md-icon"></span>
|
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8. 堆
|
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</label>
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<ul class="md-nav__list" data-md-scrollfix>
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<li class="md-nav__item">
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<a href="../../chapter_heap/heap/" class="md-nav__link">
|
|
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8.1. 堆
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</a>
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</li>
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<li class="md-nav__item">
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<a href="../../chapter_heap/build_heap/" class="md-nav__link">
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8.2. 建堆操作
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</a>
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</li>
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<li class="md-nav__item">
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<a href="../../chapter_heap/top_k/" class="md-nav__link">
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8.3. Top-K 问题（New）
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</a>
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</li>
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<li class="md-nav__item">
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<a href="../../chapter_heap/summary/" class="md-nav__link">
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8.4. 小结
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</a>
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</li>
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</ul>
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</nav>
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</li>
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<li class="md-nav__item md-nav__item--nested">
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<input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_10" >
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<div class="md-nav__link md-nav__link--index ">
|
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<a href="../../chapter_graph/">9. 图</a>
|
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|
<label for="__nav_10">
|
|
|
<span class="md-nav__icon md-icon"></span>
|
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</label>
|
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</div>
|
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|
<nav class="md-nav" data-md-level="1" aria-labelledby="__nav_10_label" aria-expanded="false">
|
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<label class="md-nav__title" for="__nav_10">
|
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<span class="md-nav__icon md-icon"></span>
|
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9. 图
|
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</label>
|
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<ul class="md-nav__list" data-md-scrollfix>
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<li class="md-nav__item">
|
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<a href="../../chapter_graph/graph/" class="md-nav__link">
|
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9.1. 图
|
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</a>
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</li>
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<li class="md-nav__item">
|
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<a href="../../chapter_graph/graph_operations/" class="md-nav__link">
|
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9.2. 图基础操作
|
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</a>
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</li>
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<li class="md-nav__item">
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<a href="../../chapter_graph/graph_traversal/" class="md-nav__link">
|
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9.3. 图的遍历
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</a>
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</li>
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<li class="md-nav__item">
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<a href="../../chapter_graph/summary/" class="md-nav__link">
|
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9.4. 小结
|
|
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</a>
|
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</li>
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</ul>
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</nav>
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</li>
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<li class="md-nav__item md-nav__item--nested">
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<input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_11" >
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<div class="md-nav__link md-nav__link--index ">
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<a href="../../chapter_searching/">10. 搜索</a>
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<label for="__nav_11">
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<span class="md-nav__icon md-icon"></span>
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</label>
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</div>
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<nav class="md-nav" data-md-level="1" aria-labelledby="__nav_11_label" aria-expanded="false">
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<label class="md-nav__title" for="__nav_11">
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<span class="md-nav__icon md-icon"></span>
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10. 搜索
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</label>
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<ul class="md-nav__list" data-md-scrollfix>
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<li class="md-nav__item">
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<a href="../../chapter_searching/binary_search/" class="md-nav__link">
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10.1. 二分查找
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</a>
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</li>
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<li class="md-nav__item">
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<a href="../../chapter_searching/binary_search_edge/" class="md-nav__link">
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10.2. 二分查找边界
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</a>
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</li>
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<li class="md-nav__item">
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<a href="../../chapter_searching/replace_linear_by_hashing/" class="md-nav__link">
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10.3. 哈希优化策略
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</a>
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</li>
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<li class="md-nav__item">
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<a href="../../chapter_searching/searching_algorithm_revisited/" class="md-nav__link">
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10.4. 重识搜索算法
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</a>
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</li>
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<li class="md-nav__item">
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<a href="../../chapter_searching/summary/" class="md-nav__link">
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10.5. 小结
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</a>
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</li>
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</ul>
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</nav>
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</li>
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<li class="md-nav__item md-nav__item--active md-nav__item--nested">
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<input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_12" checked>
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<div class="md-nav__link md-nav__link--index ">
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<a href="../">11. 排序</a>
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<label for="__nav_12">
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<span class="md-nav__icon md-icon"></span>
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</label>
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</div>
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<nav class="md-nav" data-md-level="1" aria-labelledby="__nav_12_label" aria-expanded="true">
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<label class="md-nav__title" for="__nav_12">
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<span class="md-nav__icon md-icon"></span>
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11. 排序
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</label>
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<ul class="md-nav__list" data-md-scrollfix>
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<li class="md-nav__item">
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<a href="../sorting_algorithm/" class="md-nav__link">
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11.1. 排序算法
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</a>
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</li>
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<li class="md-nav__item">
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<a href="../selection_sort/" class="md-nav__link">
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11.2. 选择排序
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</a>
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</li>
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<li class="md-nav__item">
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<a href="../bubble_sort/" class="md-nav__link">
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11.3. 冒泡排序
|
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</a>
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</li>
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<li class="md-nav__item">
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<a href="../insertion_sort/" class="md-nav__link">
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11.4. 插入排序
|
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</a>
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</li>
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<li class="md-nav__item">
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<a href="../quick_sort/" class="md-nav__link">
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11.5. 快速排序
|
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</a>
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</li>
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<li class="md-nav__item">
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<a href="../merge_sort/" class="md-nav__link">
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11.6. 归并排序
|
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</a>
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</li>
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<li class="md-nav__item">
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<a href="../heap_sort/" class="md-nav__link">
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11.7. 堆排序
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</a>
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</li>
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<li class="md-nav__item">
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<a href="../bucket_sort/" class="md-nav__link">
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11.8. 桶排序
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</a>
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</li>
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<li class="md-nav__item">
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<a href="../counting_sort/" class="md-nav__link">
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11.9. 计数排序
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</a>
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</li>
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<li class="md-nav__item">
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<a href="../radix_sort/" class="md-nav__link">
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11.10. 基数排序
|
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</a>
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</li>
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<li class="md-nav__item md-nav__item--active">
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<input class="md-nav__toggle md-toggle" type="checkbox" id="__toc">
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<label class="md-nav__link md-nav__link--active" for="__toc">
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|
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11.11. 小结
|
|
|
<span class="md-nav__icon md-icon"></span>
|
|
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</label>
|
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<a href="./" class="md-nav__link md-nav__link--active">
|
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11.11. 小结
|
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</a>
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<nav class="md-nav md-nav--secondary" aria-label="目录">
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<label class="md-nav__title" for="__toc">
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|
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<span class="md-nav__icon md-icon"></span>
|
|
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目录
|
|
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</label>
|
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<ul class="md-nav__list" data-md-component="toc" data-md-scrollfix>
|
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<li class="md-nav__item">
|
|
|
<a href="#11111-q-a" class="md-nav__link">
|
|
|
11.11.1. Q & A
|
|
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</a>
|
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</li>
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</ul>
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</nav>
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</li>
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</ul>
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</nav>
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</li>
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<li class="md-nav__item md-nav__item--nested">
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<input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_13" >
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<div class="md-nav__link md-nav__link--index ">
|
|
|
<a href="../../chapter_backtracking/">12. 回溯</a>
|
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|
|
<label for="__nav_13">
|
|
|
<span class="md-nav__icon md-icon"></span>
|
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</label>
|
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</div>
|
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<nav class="md-nav" data-md-level="1" aria-labelledby="__nav_13_label" aria-expanded="false">
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<label class="md-nav__title" for="__nav_13">
|
|
|
<span class="md-nav__icon md-icon"></span>
|
|
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12. 回溯
|
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</label>
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<ul class="md-nav__list" data-md-scrollfix>
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<li class="md-nav__item">
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<a href="../../chapter_backtracking/backtracking_algorithm/" class="md-nav__link">
|
|
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12.1. 回溯算法
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</a>
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</li>
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<li class="md-nav__item">
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<a href="../../chapter_backtracking/permutations_problem/" class="md-nav__link">
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12.2. 全排列问题
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</a>
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</li>
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<li class="md-nav__item">
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<a href="../../chapter_backtracking/subset_sum_problem/" class="md-nav__link">
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12.3. 子集和问题（New）
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</a>
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</li>
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<li class="md-nav__item">
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<a href="../../chapter_backtracking/n_queens_problem/" class="md-nav__link">
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12.4. N 皇后问题
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</a>
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</li>
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<li class="md-nav__item">
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<a href="../../chapter_backtracking/summary/" class="md-nav__link">
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|
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12.5. 小结
|
|
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</a>
|
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</li>
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</ul>
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</nav>
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</li>
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<li class="md-nav__item md-nav__item--nested">
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<input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_14" >
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<div class="md-nav__link md-nav__link--index ">
|
|
|
<a href="../../chapter_dynamic_programming/">13. 动态规划</a>
|
|
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|
|
<label for="__nav_14">
|
|
|
<span class="md-nav__icon md-icon"></span>
|
|
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</label>
|
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|
</div>
|
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|
<nav class="md-nav" data-md-level="1" aria-labelledby="__nav_14_label" aria-expanded="false">
|
|
|
<label class="md-nav__title" for="__nav_14">
|
|
|
<span class="md-nav__icon md-icon"></span>
|
|
|
13. 动态规划
|
|
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</label>
|
|
|
<ul class="md-nav__list" data-md-scrollfix>
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<li class="md-nav__item">
|
|
|
<a href="../../chapter_dynamic_programming/intro_to_dynamic_programming/" class="md-nav__link">
|
|
|
13.1. 初探动态规划（New）
|
|
|
</a>
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</li>
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<li class="md-nav__item">
|
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|
<a href="../../chapter_dynamic_programming/dp_problem_features/" class="md-nav__link">
|
|
|
13.2. DP 问题特性（New）
|
|
|
</a>
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</li>
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</ul>
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</nav>
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</li>
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<li class="md-nav__item md-nav__item--nested">
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<input class="md-nav__toggle md-toggle " type="checkbox" id="__nav_15" >
|
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<h1 id="1111">11.11. 小结<a class="headerlink" href="#1111" title="Permanent link">¶</a></h1>
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<ul>
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<li>冒泡排序通过交换相邻元素来实现排序。通过添加一个标志位来实现提前返回，我们可以将冒泡排序的最佳时间复杂度优化到 <span class="arithmatex">\(O(n)\)</span> 。</li>
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<li>插入排序每轮将未排序区间内的元素插入到已排序区间的正确位置，从而完成排序。虽然插入排序的时间复杂度为 <span class="arithmatex">\(O(n^2)\)</span> ，但由于单元操作相对较少，它在小数据量的排序任务中非常受欢迎。</li>
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<li>快速排序基于哨兵划分操作实现排序。在哨兵划分中，有可能每次都选取到最差的基准数，导致时间复杂度劣化至 <span class="arithmatex">\(O(n^2)\)</span> 。引入中位数基准数或随机基准数可以降低这种劣化的概率。尾递归方法可以有效地减少递归深度，将空间复杂度优化到 <span class="arithmatex">\(O(\log n)\)</span> 。</li>
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<li>归并排序包括划分和合并两个阶段，典型地体现了分治策略。在归并排序中，排序数组需要创建辅助数组，空间复杂度为 <span class="arithmatex">\(O(n)\)</span> ；然而排序链表的空间复杂度可以优化至 <span class="arithmatex">\(O(1)\)</span> 。</li>
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<li>桶排序包含三个步骤：数据分桶、桶内排序和合并结果。它同样体现了分治策略，适用于数据体量很大的情况。桶排序的关键在于对数据进行平均分配。</li>
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<li>计数排序是桶排序的一个特例，它通过统计数据出现的次数来实现排序。计数排序适用于数据量大但数据范围有限的情况，并且要求数据能够转换为正整数。</li>
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<li>基数排序通过逐位排序来实现数据排序，要求数据能够表示为固定位数的数字。</li>
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</ul>
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<p><img alt="排序算法对比" src="../summary.assets/sorting_algorithms_comparison.png" /></p>
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<p align="center"> Fig. 排序算法对比 </p>
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<ul>
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<li>总的来说，我们希望找到一种排序算法，具有高效率、稳定、原地以及正向自适应性等优点。然而，正如其他数据结构和算法一样，没有一种排序算法能够同时满足所有这些条件。在实际应用中，我们需要根据数据的特性来选择合适的排序算法。</li>
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</ul>
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<h2 id="11111-q-a">11.11.1. Q & A<a class="headerlink" href="#11111-q-a" title="Permanent link">¶</a></h2>
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<div class="admonition question">
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<p class="admonition-title">排序算法稳定性在什么情况下是必须的？</p>
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<p>在现实中，我们有可能是在对象的某个属性上进行排序。例如，学生有姓名和身高两个属性，我们希望实现一个多级排序/</p>
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<p>先按照姓名进行排序，得到 <code>(A, 180) (B, 185) (C, 170) (D, 170)</code> ；接下来对身高进行排序。由于排序算法不稳定，我们可能得到 <code>(D, 170) (C, 170) (A, 180) (B, 185)</code> 。</p>
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<p>可以发现，学生 D 和 C 的位置发生了交换，姓名的有序性被破坏了，而这是我们不希望看到的。</p>
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</div>
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<div class="admonition question">
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<p class="admonition-title">哨兵划分中“从右往左查找”与“从左往右查找”的顺序可以交换吗？</p>
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<p>不行，当我们以最左端元素为基准数时，必须先“从右往左查找”再“从左往右查找”。这个结论有些反直觉，我们来剖析一下原因。</p>
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<p>哨兵划分 <code>partition()</code> 的最后一步是交换 <code>nums[left]</code> 和 <code>nums[i]</code> 。完成交换后，基准数左边的元素都 <code><=</code> 基准数，<strong>这就要求最后一步交换前 <code>nums[left] >= nums[i]</code> 必须成立</strong>。假设我们先“从左往右查找”，那么如果找不到比基准数更小的元素，<strong>则会在 <code>i == j</code> 时跳出循环，此时可能 <code>nums[j] == nums[i] > nums[left]</code></strong>。也就是说，此时最后一步交换操作会把一个比基准数更大的元素交换至数组最左端，导致哨兵划分失败。</p>
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<p>举个例子，给定数组 <code>[0, 0, 0, 0, 1]</code> ，如果先“从左向右查找”，哨兵划分后数组为 <code>[1, 0, 0, 0, 0]</code> ，这个结果是不正确的。</p>
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<p>再深入思考一下，如果我们选择 <code>nums[right]</code> 为基准数，那么正好反过来，必须先“从左往右查找”。</p>
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</div>
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<div class="admonition question">
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<p class="admonition-title">关于尾递归优化，为什么选短的数组能保证递归深度不超过 <span class="arithmatex">\(log n\)</span> ？</p>
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<p>递归深度就是当前未返回的递归方法的数量。每轮哨兵划分我们将原数组划分为两个子数组。在尾递归优化后，向下递归的子数组长度最大为原数组的一半长度。假设最差情况，一直为一半长度，那么最终的递归深度就是 <span class="arithmatex">\(log n\)</span> 。</p>
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<p>回顾原始的快速排序，我们有可能会连续地递归长度较大的数组，最差情况下为 <span class="arithmatex">\(n, n - 1, n - 2, ..., 2, 1\)</span> ，从而递归深度为 <span class="arithmatex">\(n\)</span> 。尾递归优化可以避免这种情况的出现。</p>
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</div>
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<div class="admonition question">
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<p class="admonition-title">当数组中所有元素都相等时，快速排序的时间复杂度是 <span class="arithmatex">\(O(n^2)\)</span> 吗？该如何处理这种退化情况？</p>
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<p>是的。这种情况可以考虑通过哨兵划分将数组划分为三个部分：小于、等于、大于基准数。仅向下递归小于和大于的两部分。在该方法下，输入元素全部相等的数组，仅一轮哨兵划分即可完成排序。</p>
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</div>
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<div class="admonition question">
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<p class="admonition-title">桶排序的最差时间复杂度为什么是 <span class="arithmatex">\(O(n^2)\)</span> ？</p>
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<p>最差情况下，所有元素被分至同一个桶中。如果我们采用一个 <span class="arithmatex">\(O(n^2)\)</span> 算法来排序这些元素，则时间复杂度为 <span class="arithmatex">\(O(n^2)\)</span> 。</p>
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